[75-79] The great mathematician Poincaré* particularly insists on this arbitrary character of mathematics.
Of course, our mathematics fully corresponds to reality — in the sense that it is adapted to the symbolic expression of certain relations in the real; it was not, strictly speaking, impressed upon us by experience, experience merely served the mind as an occasion for its creation. But our mathematics, in the form in which it gradually developed for the convenient expression of what we needed to express, is merely one of infinitely many possible mathemat­ics or, more precisely, a particular case of some far more general mathematics, which scholars of the nineteenth century sought to attain. Having made this clear to ourselves, we immediately clarified for ourselves that mathematics, by its essence and nature, is absolutely independent of the application it receives in experience, and consequently absolutely independent of experience. It is — a free creation of the mind, the most vivid manifestation of its own creative power.

Axioms, postulates, definitions, conventions — all of this is essentially synonymous. Therefore, any conceivable mathematics could lead to conclusions which, when properly expressed by means of an appropriate system of conventions, would be entirely applicable to the real in the same way...

This theory correctly criticizes absolute rationalism and even the attenuated rationalism of Kant. It shows us that the mind was by no means obliged by inexorable necessity to develop precisely that mathematics which is so well adapted to conveying our experience; in other words, mathematics is not the expression of some universal law of reality, whether we understand reality (of course, that which is given to us) in the Cartesian manner, in the Kantian manner, or in some other way. But in Poincaré this conclusion has an entirely different meaning than in pragmatism.

Some pragmatists and even all commentators on Poincaré whom I have had occasion to read have, in my view, completely misunderstood his theory. This is an excellent example of interpretative distortion. They have made Poincaré — on this point, as on others where their error is even more profound — into an unnamed pragmatist... For the pragmatist there exists no purely contemplative and disinterested thought, there exists no pure Reason. There is only thought that wishes to master things and for this purpose distorts its representation of them in the interests of maximum convenience. Science and Reason — are servants of practice. In Poincaré, by contrast, thought is taken, so to speak, in the Aristotelian sense of the word. Thought thinks, Reason reflects for its own satisfaction; and then, beyond that, it happens that certain results of its inexhaustible creativity turn out to be useful to us for other purposes as well, besides purely spiritual satisfaction.

One may not accept Poincaré's theory in its entirety; but one should not distort it in order then to cite his authority. Insufficient attention has been paid to its connection with Kantianism,

Poincaré
Poincaré
and Kant

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